Deflection of Plumb Line

(deviation of plumb line), the angle formed by the plumb line at a given point on the earth’s surface and the normal at the same point to the surface of a certain mathematical figure that is comparable to the earth with respect to form and dimensions. In geodesy this figure is taken to be an ellipsoid of revolution, which is called the reference ellipsoid and has known dimensions and an assigned position in the body of the earth. If the deflection of the plumb line is measured in the plane containing the plumb line and the normal to the surface of the reference ellipsoid, it is called the total deflection. The total deflection of the plumb line is usually broken down into its two components, which are equal to its projections on the plane of the meridian—the deflection of the plumb in the meridian (in latitude) and, on the plane perpendicular to it, the deflection of the plumb in the prime vertical or the deflection of the plumb in longitude.

The components of the deflection of the plumb line in the meridian ξ and the prime vertical η are determined by comparing the astronomical latitude ø and longitude λ of a point on the earth’s surface with its geodetic latitude B and longitude L. The values of ξ and η are given by the formulas

ξ = ø – B η = (λ – L) cos ɸ

The component of the deflection of the plumb line in the prime vertical may also be determined by comparing the astronomical azimuth α of a certain direction with its geodetic azimuth A according to the formula

η = (α — Λ) cot ɸ

Deflections of the plumb line from the normal to the surface of the reference ellipsoid are called relative or observed deflections because they are obtained through astronomical observations and geodetic measurements. Errors of observation and measurement have a comparatively small effect on the magnitudes of relative deflections of the plumb line. They depend mainly on errors in the adopted dimensions and the assigned orientation of the reference ellipsoid in the body of the earth, as well as on irregularities in the earth’s internal structure. The magnitudes of relative deflection of the plumb line may be used to determine the deviation of the geoid from the reference ellipsoid and the dimensions and orientation of the earth’s ellipsoid that most correctly represent the figure and dimensions of the earth within the boundaries of a particular region of its surface.